Wigner–von Neumann perturbations of a periodic potential: spectral singularities in bands

Abstract

AbstractWigner–von Neumann type perturbations of the periodic one-dimensional Schrödinger operator are considered. The asymptotics of the solution to the generalized eigenfunction equation is investigated. It is proven that a subordinated solution and therefore an embedded eigenvalue may occur at the points of the absolutely continuous spectrum satisfying a certain resonance (quantization) condition between the frequencies of the perturbation, the frequency of the background potential and the corresponding quasimomentum.